MATH 412 Study Guide - Quiz Guide: Euclidean Algorithm, Division Algorithm, Linear Combination

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16 Jun 2016
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Find the greatest common divisor of 24 and 78. Write it as a linear combination of 24 and 78. If a|b and a|c, show that a divides 17b 3c. We have b = ka, k z and c = la, l z. Thus, 17b 3c = 17(ka) 3(la) which is divisible by a. Prove that every even integer is either of the form 6k or 6k + 2 or 6k + 4, for some k z. By the division algorithm, every integer is of the form 6k + r, 0 r 5. Since 6k is even, an integer 6k + r, 0 r 5 is even if and only if r = 0, 2, or 4. Compute the class of (147)(316) 122 in z15. Your answer should be of the form [a], where. [(147)(316) 122] = [ 1]7[1]6 [ 3]2 = [5] A unit is a non-zero element u such that there exists some v with uv = 1.